geodesic distance concept
| Definition | The length of the shortest path along edges between two nodes of a graph. Chapter 0 section 0.10. Also geodesic. |
|---|---|
| Example | On a circle of radius 1, two points a quarter turn apart are 1.414 apart through the plane and 1.571 apart along the circle. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id geodesic-distance, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 3.2. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | geometric-observation/chapters/ch09_legibility.md:31-41, geometric-observation/chapters/ch03_historical_precursors.md:98-110, geometric-observation/chapters/ch09_legibility.md:10-41, lean/DataMiningAsObservation/GeodesicDistance.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 3.2.
\[R(i,j)=\sum_{k\ge2}\frac1{\lambda_k}\left(\frac{u_k(i)}{\sqrt{d_i}}-\frac{u_k(j)}{\sqrt{d_j}}\right)^{2}\ \longrightarrow\ \frac1{d_i}+\frac1{d_j}\quad(n\to\infty,\ \dim\ge3).\]
Book equation 3.3.
\[X_i=r_i\,\theta_i,\qquad r_i\ \to\ \frac1{\sqrt{d_i}},\qquad \theta_i=\frac{X_i}{\|X_i\|}\in S^{m-1}.\]
Conditions
- The length of the shortest path along edges between two nodes. It is zero from a node to itself, symmetric, one between adjacent nodes, and on a connected graph obeys the triangle inequality.
- The geodesic-rank reader reads only the ordering of these distances, so a strictly increasing transform of them leaves its nearest neighbour unchanged. That the angle of the spectral embedding carries the ordering and the radius carries degree is the measured claim of chapter 3, with the refuted commute-filter row beside it.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.10 and chapter 3 section 3.3 of Data Mining
as Observation, with the geodesic-rank reader of Volume 14 chapter
9,
geometric-observation/chapters/ch09_legibility.md:10-41.
Measurements
| Where the book states it | Numbers, as the book’s sources table records them | Source |
|---|---|---|
| chapter 3 section 3.3 | row normalization is the projection onto the read subspace of the geodesic-rank consumer | geometric-observation/chapters/ch09_legibility.md:31-41;
geometric-observation/chapters/ch03_historical_precursors.md:98-110 |
| chapter 9 section 9.1 | the geodesic-rank reader discards the radius, row normalization is the angular projection | geometric-observation/chapters/ch09_legibility.md:10-41;
the-angular-observer/theorem.md:110-146 |
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/GeodesicDistance.lean,
theorems dist_self, dist_comm,
dist_triangle, dist_adj,
geodesic_rank_invariant, at observation-data-mining
f3914f0; what the check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation chapters 0, 3, 9, 10.
Related
rank-faithful; Laplacian; recognizer; bi-Lipschitz.
See also
Book equations stated beside the entry’s terms, not defining it: 0.20.
Ledger rows that cite the entry’s records without naming it: GO-3, NEG-1.
Sources-table rows that share a record with the entry without naming it: chapter 3 section 3.2, chapter 3 section 3.3, chapter 9 section 9.2, chapter 11 section 11.1.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.